2021 AMC 10A Spring 第 11 题

Question

对下列哪个整数

b

，

b

进制数

2021_b - 221_b

不能被

3

整除？

For which of the following integers

b

is the base-

b

number

2021_b - 221_b

not divisible by

3?

3

4

6

7

8

答案：E

知识点：进制整除性

难度评级：1020

小提示：

把

2021_b-221_b

换算成

10

进制。

Convert

2021_b-221_b

to base

10

大提示：

判断

2b^2(b-1)

何时能被

3

整除。

Check when

2b^2(b-1)

is divisible by

3

视频讲解：

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文字解答：

按进制定义，把这个差写成

10

进制表达式：

2b^3 + 2b + 1 - 2b^2 - 2b - 1

= 2b^3 - 2b^2 = 2b^2(b - 1)\text{。}

要使它能被

3

整除，

b

或

b - 1

必须能被

3

整除。

唯一两个条件都不满足的选项是

8

。

所以正确答案是 E。

We can express this expression in base

10

using the definition of bases:

2b^3 + 2b + 1 - 2b^2 - 2b - 1

= 2b^3 - 2b^2 = 2b^2(b - 1).

For this to be divisible by

3,

either

b

or

b - 1

must be divisible by

3.

The only answer choice that satisfies neither of these conditions is

8.

Thus, E is the correct answer.